of 1 74
A Concise Presentation Of A Theory Bridging Planetary And Atomic Scales!
By!
Ian Beardsley!
Copyright © 2025"
of 2 74
Contents
Abstract……………………………………………………….3!
Part 1 The Theory
Concise Presentation……………………………………..…5!
Jupiter And Saturn…………………………………………………….11
The Solar Formulation………………………………………13!
Equating The Lunar And Solar
Formulations Yield Our 1 Second Base Unit…………………16
Part 2 Modeling With A Computer Program
Ideal Way to Model Habitable Star Systems…………….19!
Finding The Smooth Curves…………………………….…28!
A Theory For Biological Hydrocarbons…………………..58!
Lunar Eclipses………………………………………………63!
The Archaeological Component…………………………..69!
Appendix 1…………………………………………………..71!
Appendix 2…………………………………………………..73!
of 3 74
Abstract
It is found the planetary system, our Solar System, and atomic systems, can be described in
terms of the same characteristic time. That time turns out to be 1 second, which happens to be
the fundamental unit we use to measure time. That system comes from the ancient Sumerians
using base 12 and base 60 counting, ultimately, but in light of its appearance in natural
structures in our theory with such recurrence, it would seem it is not an arbitrary unit, but has
natural significance. This theory, involving a wave equation solution to the Solar System
provides another way to model the formation of star systems other than using standard
celestial mechanics. Here we model with a computer program, after presenting the theory in
Part 1, star systems from spectral types F5V (blue stars) through G type stars like our Sun
(yellow stars), to K3V stars (orange stars). It is suggested that the same characteristic time of
one second that describes planetary and atomic systems describes hydrocarbons, the
chemical skeletons of life chemistry, connecting our physical theory to biological theory. At the
end of the paper the Moon eclipsing the Sun is treated over geologic time."
of 4 74
Part 1 The Theory!
of 5 74
Concise Presentation In the Bohr atom the radius of the electron’s orbit in the ground state
(n=1) is given by!
1. !
2. !
I have extended this to our solar system and found that the ground state of our solar system is
based on the Moon of our Earth. I have written!
3. !
Yes, interestingly it is equal to 1 second, our base unit of time that ultimately came from the
way the Ancient Sumerians divided up the period of rotation of the Earth (1 Day) into 24 hours,
each hour into 60 minutes, each minute into 60 seconds. They used based 60 counting which
was handed down to the Babylonians. is the mass of the Moon, is the Planck-type
constant I have proposed for the the Solar System. I found it is given by!
4. !
Where is the kinetic energy of the Earth in its orbit around the Sun. We see the the third
planet from the Sun, the Earth, determines its Planck constant in terms of the same unit of a
second. I also find that the kinetic energy of the Moon to the kinetic energy of the Earth, that
quantity times the Earth rotation period (24 hours, its day) is close to a second, about 1.2 to 1.3
seconds depending whether you use aphelion or perihelions, and in what combination. That is!
5. !
If the second were purely arbitrary, we should expect this ratio to not be so close to 1 second. I
also find that the proton has a characteristic time of 1 second as well, in two forms that when
equated give about the radius of the proton:!
6. !
7. !
Equating these two yield the radius of a proton, :!
r
1
=
ℏ
2
ke
2
m
e
R
1
≈ 0.529E − 10m
ℏ
2
⊙
GM
3
m
⋅
1
c
= 1second
M
m
ℏ
⊙
ℏ
⊙
= (1second )KE
e
KE
e
KE
m
KE
e
(EarthDay) = 1.2 − 1.3seconds
(
1
6α
2
4πh
Gc
)
⋅
r
p
m
p
= 1second
2
3
⋅
πr
p
α
4
Gm
3
p
1
3
⋅
h
c
= 1second
r
p
of 6 74
8. !
I find if I replace 2/3 with , we may have exactly the radius of a proton!
9. !
This may show an underlying optimization principle where occurs in biological and cosmic
structures. I find I can arrive at this radius of a proton another way, With the Planck energy,
, given by frequency of a particle, and from mass-energy equivalence, :!
We take the rest energy of the mass of a proton :
The frequency of a proton is
Since our theory gave us the factor of 2/3 for the radius of a proton we have:
The radius of a proton is then
This is very close to the value upon which the proton radius converged historically by two
independent methods which was 0.877E-15m. The result from our theory is
The 0.877fm was challenged in 2010 by a third experiment making it 4% smaller and was
0.842E-15m. We find it may be that the radius of a proton is actually
r
p
=
2
3
⋅
h
cm
p
ϕ = 1.618
r
p
= ϕ
h
cm
p
ϕ
E = h f
E = mc
2
E = h f
m
p
E = m
p
c
2
f
p
=
m
p
c
2
h
m
p
c
2
h
⋅
r
p
c
=
2
3
≈ ϕ =
m
p
c
h
r
p
m
p
r
p
=
2
3
⋅
h
c
r
p
=
2
3
⋅
h
cm
p
r
p
=
2
3
⋅
6.62607E − 34
(299,792,458)(1.67262E − 27)
= 0.88094E − 15m
r
p
= ϕ ⋅
h
cm
p
= 0.816632E − 15m
of 7 74
The theory becomes archaeological. I guessed since one second comes from the ancient
Sumerians dividing the Earth day (rotation period) into 24 hours, and those into 60 minutes,
and those into 60 seconds, that that has to do with the rotational angular momentum of the
Earth, . I found
10.
Where
11.
The value is 2.5 which by modeling our Solar System is found to be the exponent in the pressure
gradient for the protoplanetary disc from which our Solar System formed. That is I found
12.
Which suggests that the structure of the protoplanetary disc could be governed by the same
fundamental time of one second in the Earth’s rotation and that the Earth’s formation process
may be encoded in the same number we developed since ancient times to describe time (24, 60).
This is the solution to:
13.
The protoplanetary disc that evolves into the planets has two forces that balance its pressure, the
centripetal force of the gas disc due to its rotation around the protostar and the inward
gravitational force on the disc from the protostar , and these are related by the density
of the gas that makes up the disc. It is the pressure gradient of the disc in radial equilibrium
balancing the inward gravity and outward centripetal force. In order to apply this to other star
systems, we have to be able to predict the radius of the habitable planet, presumably in the n=3
orbit. I found the answer to be in the Vedic literature of India. They noticed that the diameter of
the Sun is about 108 times the diameter of the Earth and that the average distance from the Sun
to the Earth is about 108 solar diameters, with 108 being a significant number in Yoga. So I
wrote:
14.
The surprising result I found was, after applying it to the stars of many spectral types, with their
different radii and luminosities (the luminosities determine , the distances to the habitable
zones) that the radius of the planet always came out about the same, about the radius of the
Earth. This may suggest optimally habitable planets are not just a function of the distance from
the star, which determines their temperature, but are functions of their size and gravity. In order
L
earth
L
earth
ℏ
⊙
24 = 60
L
earth
=
4
5
π M
e
f
e
r
2
e
P(R) = P
0
(
R
R
0
)
−
L
earth
ℏ
⊙
d P
dr
= − ρ
(
GM
⋆
r
2
−
v
2
ϕ
r
)
v
2
ϕ
/r
GM
⋆
/r
2
ρ
R
planet
= 2
R
2
⋆
r
planet
r
planet
of 8 74
to get , the distance of the habitable planet from the star, we use the inverse square law for
luminosity of the star. If the Earth is in the habitable zone, and if the star is one hundred times
brighter than the Sun, then by the inverse square law the distance to the habitable zone of the
planet is 10 times that of what the Earth is from the Sun. Thus we have in astronomical units the
habitable zone of a star is given by:
15.
Also, the theory utilizes the fact that the Moon as seen from the Earth perfectly eclipses the Sun
as a possible condition for optimal habitability of the planet, which is
16.
The Schrödinger Wave Equation must be solved to determine the energies and orbitals of the
electron in the hydrogen atom. In spherical coordinates it is
17.
It has the solutions
18.
19.
I find the solutions are for the Earth orbiting the Sun are:
20.
21.
To solve the wave equation in the case of a protoplanetary disc you would want the wave
equation to be in cylindrical coordinates, to solve it looks like this
The time independent Schrödinger equation is
22.
r
planet
r
planet
=
L
⋆
L
⊙
AU
r
planet
r
moon
=
R
star
R
moon
−
ℏ
2
2m
[
1
r
2
∂
∂r
(
r
2
∂
∂r
)
+
1
r
2
sinθ
∂
∂θ
(
sinθ
∂
∂θ
)
+
1
r
2
sin
2
θ
∂
2
∂ϕ
2
]
ψ + V(r)ψ = E ψ
E
n
= −
Z
2
(k
e
e
2
)
2
m
e
2ℏ
2
n
2
r
n
=
n
2
ℏ
2
Z k
e
e
2
m
e
K E
e
= n
R
⊙
R
m
⋅
G
2
M
2
e
M
3
m
2ℏ
2
⊙
r
n
=
2ℏ
2
⊙
GM
3
m
⋅
R
⊙
R
m
⋅
1
n
−
ℏ
2
2M
e
∇
2
Ψ(ρ, ϕ, z) + V(ρ, z)Ψ(ρ, ϕ, z) = E Ψ(ρ, ϕ, z)
of 9 74
The Laplacian in cylindrical coordinates is
23.
The gravitational potential is
24.
Where is the mass of the central protostar. We can refer to the variables as such:
We separate the variables according to:
25.
Substituting 23 and 25 into 1 we have
26.
Solving it all the way is beyond the scope of this paper. We have hindsight, so we were able to
write out the solutions by form and dimensional analysis. Let us now compute the value of
and show that the units work…
∇
2
=
1
ρ
∂
∂ρ
(
ρ
∂
∂ρ
)
+
1
ρ
2
∂
2
∂ϕ
2
+
∂
2
∂z
2
V(ρ , z) = −
GM
s
ρ
2
+ z
2
M
s
Ψ(ρ, ϕ, z) = R(ρ) ⋅ Φ(ϕ) ⋅ Z(z)
−
ℏ
2
2M
e
[
1
ρ
d
dρ
(
ρ
d R
dρ
)
Φ ⋅ Z +
1
ρ
2
R ⋅
d
2
Φ
dϕ
2
⋅ Z + R ⋅ Φ ⋅
d
2
Z
d z
2
]
+ V(ρ, z)RΦZ = ERΦZ
ℏ
⊙
of 10 74
Where
Where is the radius of a proton, is the mass of a proton, is the speed of light, and is the
fine structure constant. I found this gives the characteristic time of one second in terms of a
proton (Equation 1.7). We guess the planetary scale is connected to the proton scale because the
planets formed from the protoplanetary disc and are made of different combinations of protons.
We derive the value of our solar Planck constant
=
=
=
=
27.
ℏ
⊙
= (hC )K E
e
hC = 1second
C =
1
3
⋅
1
α
2
c
2
3
⋅
π r
p
G m
3
p
r
p
m
p
c
α
C =
1
3
⋅
1
α
2
c
1
3
⋅
2π r
p
G m
3
p
1
3
⋅
18769
299792458
1
3
⋅
2π (0.833E − 15)
(6.67408E − 11)(1.67262E − 27)
3
1.55976565E 33
s
m
m
kg
3
⋅
s
2
kg
m
3
=
s
m
s
2
kg
2
m
2
=
s
m
⋅
s
kg ⋅ m
=
1
kg
⋅
s
2
m
2
1
C
= kg
m
2
s
2
=
1
2
mv
2
= energ y
hC = (6.62607E − 34)(1.55976565E 33) = 1.03351secon d s ≈ 1.0secon d s
hC =
(
kg
m
s
2
⋅ m ⋅ s
)
(
1
kg
⋅
s
2
m
2
)
(
kg
m
2
s
)(
1
kg
⋅
s
2
m
2
)
= seconds
K E
earth
=
1
2
(5.972E 24kg)(30,290m /s)
2
= 2.7396E 33J
ℏ
⊙
= (hC )K E
earth
= (1.03351s)(2.7396E 33J ) = 2.8314E 33J ⋅ s
of 11 74
Jupiter And Saturn Now we want to find what the wave equation solutions are for Jupiter
and Saturn because they significantly carry the majority of the mass of the solar system and thus
should embody most clearly the dynamics of the wave solution to the Solar System. We also also
show here how well the solution for the Earth works, which 99.5% accuracy.
I find that as we cross the asteroid belt leaving behind the terrestrial planets, which are solid,
and go to the gas giants and ice giants, the atomic number is no longer squared and the square
root of the the orbital number moves from the numerator to the denominator. I believe this is
because the solar system here should be modeled in two parts, just as it is in theories of solar
system formation because there is a force other than just gravity of the Sun at work, which is the
radiation pressure of the Sun, which is what separates it into two parts, the terrestrial planets on
this side of the asteroid belt and the giants on the other side of the asteroid belt. The effect the
radiation pressure has is to blow the lighter elements out beyond the asteroid belt when the
solar system forms, which are gases such as hydrogen and helium, while the heavier elements
are too heavy to be blown out from the inside of the asteroid belt, allowing for the formation of
the terrestrial planets Venus, Earth, and Mars. The result is that our equation has the atomic
number of the heavier metals such as calcium for the Earth, while the equation for the giants has
the atomic numbers of the gasses. We write for these planets
So, for Jupiter we have (And again using the maximum orbital velocity which is at perihelion):
Jupiter is mostly composed of hydrogen gas, and secondly helium gas, so it is appropriate that
.
Our equation for Jupiter is
28.
Where is the atomic number of hydrogen which is 1 proton, and for the orbital
number of Jupiter, . Now we move on to Saturn…
E =
Z
n
G
2
M
2
m
3
2ℏ
2
⊙
K E
j
=
1
2
(1.89813E 27kg)(13720m /s)
2
= 1.7865E 35J
E =
Z
H
5
(6.67408E − 11)
2
(1.89813E 27kg)
2
(7.347673E 22kg)
3
2(2.8314E 33)
2
E =
Z
H
5
(3.971E 35J ) = Z
H
(1.776E 35J )
Z
H
=
1.7865E 35J
1.776E 35J
= 1.006proton s ≈ 1.0proton s = hydr ogen(H )
Z = Z
H
E
5
=
Z
H
5
G
2
M
2
j
M
3
m
2ℏ
2
⊙
Z
H
n = 5
n = 5
of 12 74
=
The equation for Saturn is then
29.
It makes sense that Saturn would use Helium in the equation because Saturn is the next planet
after Jupiter and Jupiter uses hydrogen, and helium is the next element after hydrogen. As well,
just like Jupiter, Saturn is primarily composed of hydrogen and helium gas.
The accuracy for Earth orbit is!
!
=
=2.727E36J
The kinetic energy of the Earth is
Which is very good, about 100% for all practical purposes. The elemental expression of the
solution for the Earth would be
30.
K E
S
=
1
2
(5.683E 26kg)(10140m /s)
2
= 2.92E 34J
E =
Z
6
(6.67408E − 11)
2
(5.683E 26kg)
2
(7.347673E 22)
3
2(2.8314E 33)
2
Z
2.45
(3.5588E 34J ) = Z(1.45259E 34J )
Z(1.45259E 34J ) = (2.92E 34J )
Z = 2proton s = Helium(He)
E
6
=
Z
He
6
G
2
M
2
s
M
3
m
2ℏ
2
⊙
E
n
= n
R
⊙
R
m
G
2
M
2
e
M
3
m
2ℏ
2
⊙
R
⊙
R
m
=
6.96E8m
1737400m
= 400.5986
E
3
= (1.732)(400.5986)
(6.67408E − 11)
2
(5.972E 24kg)
2
(7.347673E 22kg)
3
2(2.8314E 33)
2
K E
e
=
1
2
(5.972E 24kg)(30,290m /s)
2
= 2.7396E 33J
2.727E 36J
2.7396E 33J
100 = 99.5 %
E
3
= 3
Z
2
Ca
G
2
M
2
e
M
3
m
2ℏ
2
⊙
of 13 74
Where
In this case the element associated with the Earth is calcium which is Z=20 protons.
The Solar Formulation Our solution of the wave equation for the planets gives the kinetic
energy of the Earth from the mass of the Moon orbiting the Earth, but you could formulate
based on the Earth orbiting the Sun. In our lunar formulation we had:
31.
We remember the Moon perfectly eclipses the Sun which is to say
32.
Thus equation 31 becomes
33.
The kinetic energy of the Earth is
34.
Putting this in equation 33 gives the mass of the Sun:
35.
We recognize that the orbital velocity of the Moon is
36.
So equation 35 becomes
37.
This gives the mass of the Moon is
R
⊙
R
m
→ Z
2
K E
e
= 3
R
⊙
R
m
⋅
G
2
M
2
e
M
3
m
2ℏ
2
⊙
R
⊙
R
m
=
r
e
r
m
K E
e
= 3
r
e
r
m
⋅
G
2
M
2
e
M
3
m
2ℏ
2
⊙
K E
e
=
1
2
⋅
GM
⊙
M
e
r
e
M
⊙
= 3r
2
e
⋅
GM
e
r
m
⋅
M
3
m
ℏ
2
⊙
v
2
m
=
GM
e
r
m
M
⊙
= 3r
2
e
⋅ v
2
m
⋅
M
3
m
ℏ
2
⊙
of 14 74
38.
Putting this in equation 31 yields
39.
We now multiply through by and we have
40.
Thus the Planck constant for the Sun, , in this the case the star is the Sun, is angular
momentum quantized, the angular momentum we will call , the subscript for Planck. We
have
We write for the solution of the Earth/Sun system:
41.
We can write 41 as
42.
We say. . That is
Let us see how accurate our equation is:
M
3
m
=
M
⊙
ℏ
2
⊙
3r
2
e
v
2
m
K E
e
=
R
⊙
R
m
⋅
G
2
M
2
e
M
⊙
2r
2
e
v
2
m
M
2
e
/M
2
e
K E
e
=
R
⊙
R
m
⋅
G
2
M
4
e
M
⊙
2r
2
e
v
2
m
M
2
e
ℏ
⊙
L
p
p
L
p
= r
e
v
m
M
e
= r
e
v
m
M
e
= (1.496E11m)(1022m /s)(5.972E 24kg) = 9.13E 38kg ⋅
m
2
s
L
2
p
= r
2
e
v
2
m
M
2
e
= 7.4483E 77J ⋅ m
2
⋅ kg = 8.3367E 77kg
2
⋅
m
4
s
2
K E
e
=
R
⊙
R
m
⋅
G
2
M
4
e
M
⊙
2L
2
p
K E
e
=
R
⊙
R
m
⋅
G
2
M
4
e
M
⊙
2ℏ
2
⊙
ℏ
⊙
= ℏ
⊙
/2π
ℏ
⊙
= 9.13E 38J ⋅ s
h
⊙
= 2π ℏ
⊙
= 5.7365E 39J ⋅ s
K E
e
=
R
⊙
R
m
⋅
G
2
M
4
e
M
⊙
2ℏ
2
⊙
of 15 74
=
=
We have that the kinetic energy of the Earth is
Our equation has an accuracy of
Which is very good.
We call the same in our solar solution. But we now want our solution for the solar
formulation. is the mass of the Earth and is the mass of the Sun. We have our solution
might be!
43. !
Where is !
We have
44.
This has an accuracy of
!
Thus the solutions to the wave equation!
R
⊙
R
m
(6.67408E − 11)
2
(5.972E 24kg)
4
(1.9891E 30kg)
2(8.3367E 77kg
2
⋅
m
4
s
2
)
R
⊙
R
m
(6.759E 30J )
R
⊙
R
m
=
6.957E8m
1737400m
= 400.426
K E
e
= 2.70655E 33J
K E
earth
=
1
2
(5.972E 24kg)(30,290m /s)
2
= 2.7396E 33J
2.70655E 33J
2.7396E 33J
= 98.79 %
L
p
ℏ
⊙
r
n
M
e
M
⊙
r
n
=
ℏ
2
⊙
GM
3
e
⋅
R
m
R
⊙
h
⊙
ℏ
⊙
= 9.13E 38J ⋅ s
r
3
=
(9.13E 38)
2
(6.67408E − 11)(5.972E 24kg)
3
⋅
1
400.5986
= 1.4638E11m
1.4638E11m
1.496E11m
100 = 97.85 %
of 16 74
!
for the solar formulations are!
45.
46.
Equating The Lunar And Solar Formulations Yield Our 1 Second Base Unit
Let us equate equations with equations:
47.
48.
This gives:
49.
We remember that
−
ℏ
2
2m
[
1
r
2
∂
∂r
(
r
2
∂
∂r
)
+
1
r
2
sinθ
∂
∂θ
(
sinθ
∂
∂θ
)
+
1
r
2
sin
2
θ
∂
2
∂ϕ
2
]
ψ + V(r)ψ = E ψ
K E
e
=
R
⊙
R
m
⋅
G
2
M
4
e
M
⊙
2ℏ
2
⊙
r
n
=
ℏ
2
⊙
GM
3
e
⋅
R
m
R
⊙
ℏ
⊙
= 9.13E 38J ⋅ s
h
⊙
= 2π ℏ
⊙
= 5.7365E 39J ⋅ s
K E
e
= n
R
⊙
R
m
⋅
G
2
M
2
e
M
3
m
2ℏ
2
⊙
K E
e
=
R
⊙
R
m
⋅
G
2
M
4
e
M
⊙
2ℏ
2
⊙
3
R
⊙
R
m
⋅
G
2
M
2
e
M
3
m
2ℏ
2
⊙
=
R
⊙
R
m
⋅
G
2
M
4
e
M
⊙
2L
2
p
L
p
=
M
2
e
M
⊙
M
3
m
3
⋅ ℏ
⊙
ℏ
⊙
= (hC )K E
p
hC = 1second
C =
1
3
⋅
1
α
2
c
1
3
⋅
2π r
p
G m
3
p
of 17 74
This gives
50.
We have
51.
This equates the orbital velocity of the Moon with the centripetal acceleration of the Earth in
terms of one second by way of the mass of the Earth, the mass of the Sun, the mass of the Moon,
and the orbital number of the Earth. Let us compute
52.
Let us see how well equation 51 works. at aphelion is 966 m/s and .
. We have
That is an accuracy of
Equation 51 can be written:
53.
In terms of other star systems this is
54.
1
6α
2
m
p
h 4π r
2
p
Gc
= 1.004996352secon ds
r
e
v
m
M
e
=
1
6α
2
⋅
r
p
m
p
h 4π
Gc
⋅
1
2
M
e
v
2
e
⋅
M
2
e
M
⊙
M
3
m
3
2v
m
=
v
2
e
r
e
(1secon d )
M
2
e
M
⊙
M
3
m
3
M
2
e
M
⊙
M
3
m
3
=
(5.972E 24kg)
2
(1.9891E 30kg)
(7.34763E 22kg)
3
(1.732)
= 321,331.459 ≈ 321,331
v
m
v
e
= 29,800m /s
r
e
= 1AU = 1.496E11m
2(966m /s) =
(29,800m /s)
2
1.496E11m
(1sec)(321,331.459)
1,907m /s = 1,932m /s
1907
1932
= 98.7 %
1secon d = 2r
e
v
m
v
2
e
M
3
m
3
M
2
e
M
⊙
1secon d = 2r
p
v
m
v
2
p
M
3
m
3
M
2
p
M
⋆
of 18 74
Part 2 Modeling With A Computer Program!
of 19 74
Ideal Way to Model Habitable Star Systems The best way to solve star systems with our
theory would be to use!
1.0 , 3.0 !
2.0. , 4.0. !
Where here . Equation 2.0 becomes!
5.0. !
Where for Earth p=2.5, the exponent in the pressure gradient for its protoplanetary disc. From
this we get . We now get the characteristic time, , from!
!
by using!
6.0. and 7.0. !
And we have . We can now put in equation 1.0!
!
Which here is 1 second for the Earth, to get the mass of the moon, . But to use equation 5,
we need from equation 3.0. This requires the mass of the Earth, the frequency of the
earth, which we get from the planet’s day (Its rotation rate) and the radius of the planet. We
have all of these values for habitable planets in a K2V star and an M2V star, but they are tidal
locked. We have the frequencies because the planets are tidally locked, so their planets
rotation periods are equal to their orbital periods. If we can’t measure the planet’s radius in
another star system, we might obtain it from: !
8.0. !
Which works for the Earth. We can get the orbital radius of our Moon from!
ℏ
2
⊙
GM
3
m
⋅
1
c
= 1second
L
earth
=
4
5
πM
e
f
e
R
2
e
P(R) = P
0
(
R
R
0
)
−L
ear th
ℏ
⊙
ℏ
⊙
= t
c
KE
e
t
c
= 1second
L
earth
ℏ
⊙
= p
ℏ
⊙
t
c
ℏ
⊙
= t
c
KE
e
v
e
=
GM
e
r
e
KE
earth
=
1
2
M
e
v
2
e
t
c
t
c
ℏ
2
⊙
GM
3
m
⋅
1
c
= 1second
M
m
L
earth
R
e
=
2R
2
⊙
r
e
of 20 74
9.0. !
The radius of the planet’s moon we suggested is given by a perfect eclipse:!
10.0. !
Modeling Our Solar System***************************************************************************!
We have already applied our theory to the Earth/Moon/Sun System and it worked out nicely.
Now we want to apply this ideal approach we just outlined, to it, so we can test it. We start with
the angular momentum of the Earth. It is given by!
Or,…!
!
= !
The orbital velocities and kinetic energies of the Earth are given by:!
!
!
We can now determine :!
!
This is correct for our solar system’s Planck constant. We have the characteristic time is!
!
Which is correct as well. Now we compute the mass of our moon…!
!
r
m
= R
⊙
Ag
Au
=
R
⋆
(1.8)
R
⋆
R
m
=
r
p
r
m
L
earth
=
4
5
πM
e
f
e
R
2
e
L
e
=
4
5
π(5.972E 24kg)
1
(86400seconds)
(6.378E6m)
2
7.07866672E 33J ⋅ s
v
p
=
GM
⋆
r
p
=
(6.674E − 11)(1.989E30kg)
(1.496E11m)
= 29,788.24m /s
KE
p
=
1
2
M
p
v
2
p
=
1
2
(5.972E 24kg)(29,788.24m /s)
2
= 2.65E33J
ℏ
⋆
ℏ
⋆
=
L
p
p
=
(7.07866672E 33J ⋅ s)
2.5
= 2.831467E33J ⋅ s
t
c
=
ℏ
⋆
KE
p
=
(2.831467E33J ⋅ s)
2.65E33J
= 1.068seconds ≈ 1second
M
3
m
=
(2.831467E33J ⋅ s)
2
(6.674E − 11)(299,729,458m /s)(1.068secon ds)
of 21 74
!
This is also very accurate (actual value: 7.347673kg. Now we compute the orbital radius of the
Moon…!
!
This is accurate too (actual value: 3.84E8m). From this we have the radius of the Moon:!
!
This is pretty accurate, too. The actual value is 1.7374E6m!
Now to get the density of the Moon…!
!
!
This is good, our Moon is about 3.344g/cm3. Now we want to check!
!
!
!
So this gives the correct characteristic time for the Earth/Moon/Sun system. Let’s compute the
planet day characteristic time…!
!
!
M
m
= 7.213E 22kg
r
m
= R
⋆
Ag
Au
= R
⋆
/(1.8) =
6.957E8m
1.8
= 3.865EE8m
R
m
= R
⋆
r
m
r
p
= (6.957E8m)
3.865E8m
1.496E11m
= 1.79738E6m
V
m
=
4
3
πR
3
m
=
4
3
π(1.79738E6m)
3
= 2.432E19m
3
ρ
m
=
7.213E 22kg
2.432E19m
3
= 2.96587g/cm
3
≈ 3g/cm
3
1second = 2r
e
v
m
v
2
e
M
3
m
3
M
2
e
M
⊙
v
m
=
GM
p
r
m
=
(6.674E − 11)(5.972E 24kg)
(3.865EE8m)
= 1,015.5m /s
2(1.496E11m)
1,015.5m /s
(29,788.24m /s)
2
(7.213E 22kg)
3
(1.732)
(5.972E 24kg)
2
(1.989E30kg)
= 1.03648sec ≈ 1secon d
1second ≈
KE
m
KE
e
(EarthDay)
KE
m
=
1
2
(7.213E 22kg)(1,015.5m /s)
2
= 3.719E 28J
of 22 74
!
We see the system for modeling star system works. The thing is while M class stars are the
most abundant in the galaxy and have longer life spans than the Sun, their planets are thought
to be tidally locked, their day is equal to their year, leaving them perpetually night on one side
of the planet and perpetually day on the other, only being cool enough for life in the twilight
region between night and day. K class stars show a lot of promise to host life on planets in their
habitable zones because they are far enough away from their star that they might not always
become tidally locked, while being more stable than the Sun and longer lived. It is easiest to
detect planets and get data for these M2V and K2V stars, but when you get to our Sun a G2V
star, they are so bright they wash out the light of their planets in the habitable zone. However,
we have gotten data for a planet in the habitable zone of a G2V star, the same size, radius,
mass, and luminosity as our Sun. And the planet in its habitable zone has the same radius, and
mass as our Earth. All we need to do is to detect a moon around its planet, and we will have
verified my theory for G2V stars. This star is called KOI-4878 and its planet is KOI-4878.01.
Here is the information on it…!
!
For ease of modeling and checking my computations I have made a program in C that models
according to this ideal method. The round function has not been incorporated so the program
truncates after 8 digits. I only work to about 4 digits after the decimal so it doesn’t seem to be
much of a problem. You will see all of the program output verifies my computations, and as well
in much of the in-between work, not just the final results. Here is the program and running it for
several spectral classes (You can see the formula for the radius of the planet always comes out to
be around that of the Earth)…
KE
m
KE
e
(PlanetDay) =
(3.719E 28J )
(2.65E33J )
(86,400sec) = 1.2seconds
of 23 74
F8V Star
Mass: 1.18
Radius: 1.221
Luminosity: 1.95
F9V Star
Mass: 1.13
Radius: 1.167
Luminosity: 1.66
G0V Star
Mass: 1.06
Radius: 1.100
Luminosity: 1.35
G1V Star
Mass: 1.03
Radius: 1.060
Luminosity: 1.20
As you can see we consistently get about 1 Earth radius for the radius of every planet in the
habitable zone of each type of star. I have gone through all stars down to G8V and consistency
got this result. You will see these results when I run the program for the stars from F8V to G8V
M
⋆
= 1.18(1.9891E 30kg) = 2.347E 30kg
R
⋆
= 1.221(6.9634E8m) = 8.5023E 8m
r
p
= 1.95L
⊙
AU = 1.3964AU(1.496E11m /AU ) = 2.08905E11m
R
p
=
2R
2
⋆
r
p
= 2
(8.5023E8m)
2
2.08905E11m
=
6.92076E6m
6.378E6m
= 1.0851Ear th Ra dii
M
⋆
= 1.13(1.9891E 30kg) = 2.247683E 30kg
R
⋆
= 1.167(6.9634E8m) = 8.1262878E 8m
r
p
= 1.66L
⊙
AU = 1.28841AU(1.496E11m /AU ) = 1.92746E11m
R
p
=
2R
2
⋆
r
p
= 2
(8.1262878E8m)
2
1.92746E11m
=
6.852184E6m
6.378E6m
= 1.0743468Ear th Ra dii
M
⋆
= 1.06(1.9891E 30kg) = 2.108446E 30kg
R
⋆
= 1.100(6.9634E8m) = 7.65974E8m
r
p
= 1.35L
⊙
AU = 1.161895AU(1.496E11m /AU ) = 1.7382E11m
R
p
=
2R
2
⋆
r
p
= 2
7.65974E8m)
2
1.7382E11m
=
6.751E6m
6.378E6m
= 1.05848Ear th Ra dii
M
⋆
= 1.03(1.9891E 30kg) = 2.11E 30kg
R
⋆
= 1.060(6.9634E8m) = 7.381E8m
r
p
= 1.20L
⊙
AU = 1.0954AU(1.496E11m /AU ) = 1.63878589E11m
R
p
=
2R
2
⋆
r
p
= 2
7.3812E8m)
2
1.63878589E11m
=
6.6491E6m
6.378E6m
= 1.0425Ear th Ra dii
of 24 74
//
// main.c
// Intersections
//
// Created by Ian Beardsley on 1/7/25.
//
#include <stdio.h>
#include <math.h>
int main(int argc, const char * argv[]) {
float R_p, M_p, R_s, M_s, L_s, h_s, t_c, M_m, rho_m, rho_p, PlanetDay,
V_p,StarRadius, PlanetRadius, PlanetMass, StarLuminosity, PlanetOrbit,
StarMass, r_p, T_p, p, L_p, KE_p, v_p, Root;
float G=6.674E-11, hbarstar, PDCT;
float Mcubed, r_m, root, civilization, R_m, V_m, MoonDensity, part1,
part2, part3,v_m, KE_m;
int i;
printf ("What is the radius of the star in solar radii? ");
scanf ("%f", &StarRadius);
printf ("What is the mass of the star in solar masses? ");
scanf ("%f", &StarMass);
printf ("What is the luminosity of the star in solar luminosities? ");
scanf ("%f", &StarLuminosity);
PlanetOrbit=sqrt(StarLuminosity);
r_p=PlanetOrbit*1.496E11;
printf ("What is the radius of the planet in Earth radii? May we suggest
it is given by 2StarRadius^2/PlanetOrbitalRadius? 1=yes, 0=no ");
scanf ("%i", &i);
R_p=0;
if (i==1)
{
R_s=6.9634E8*StarRadius;
R_p=(2*R_s*R_s)/r_p;
printf ("The radius of the planet is: %fE6m\n", R_p/1E6);
printf ("that is %f Earth Radii \n", R_p/6.378E6);
}
else
{
printf("What is the planet radius in Earth radii?: ");
scanf("%f", &PlanetRadius);
R_p=PlanetRadius*6.378E6;
}
printf("What is the mass of the planet in Earth masses? May we
suggest that it is such that it has the density of the Earth? 1=yes, 0=no ");
scanf("%i", &i);
if (i==1)
{
V_p=1.33333*(3.14159)*R_p*R_p*R_p;
M_p=V_p*5510;
PlanetMass=M_p/5.972E24;
of 25 74
printf("The mass of the planet is %f E24kg \n", M_p/1E24);
printf("That is %f Earth masses \n", PlanetMass);
}
else
{
printf("What is the mass of the planet in Earth Masses? ");
scanf("%f", &PlanetMass);
M_p=PlanetMass*5.972E24;
printf("The mass of the Planet is %f E24kg \n", M_p/1E24);
printf ("What is the planet day in Earth days? ");
scanf ("%f", &PlanetDay);
T_p=PlanetDay*86400;
printf("That is %f seconds \n", T_p);
{
printf("What is p the pressure gradient exponent of the
protoplanetary disc? ");
scanf("%f", &p);
printf("What integer do you want to use, F0V=2,G0V=1? ");
scanf("%f", &root);
printf("Choose copper-silver civilization (1.7) or gold silver
(1.8)? ");
scanf("%f", &civilization);
Root=sqrt(root);
M_s=1.9891E30*StarMass;
v_p=sqrt(G*M_s/r_p);
L_p=0.8*3.14159*M_p*(1/T_p)*R_p*R_p;
KE_p=0.5*M_p*v_p*v_p;
hbarstar=L_p/p;
t_c=hbarstar/KE_p;
part1=cbrt(hbarstar/(t_c));
part2=cbrt(1/G);
part3=cbrt(hbarstar/299792458);
M_m=part1*part2*part3;
R_s=StarRadius*6.9634E8;
r_m=R_s/(Root*civilization);
R_m=R_s*r_m/r_p;
V_m=1.33333*3.14159*R_m*R_m*R_m;
rho_m=(M_m/V_m);
MoonDensity=rho_m*0.001;
V_p=1.33333*3.14159*R_p*R_p*R_p;
rho_p=(M_p/V_p)*0.001;
printf("\n");
printf("\n");
printf("Angular Momentum of Planet: %f E33 \n", L_p/1E33);
of 26 74
printf("\n");
printf("\n");
printf("planet orbital velocity: %f m/s \n", v_p);
printf("planet mass: %f E24 kg \n", M_p/1E24);
printf("planet mass: %f Earth masses \n", M_p/5.972E24);
printf("planet orbital radius: %f E11 m \n", r_p/1E11);
printf ("planet orbital radius: %f Earth distances \n", r_p/
1.496E11);
printf("planet KE: %f E33 J \n",KE_p/1E33);
printf("planet density: %f g/cm3 \n", rho_p);
printf("\n");
printf("\n");
printf("hbarstar: %f E33 Js \n", hbarstar/1E33);
printf("characteristic time: %f seconds\n", t_c);
printf("\n");
printf("\n");
printf("Orbital Radius of Moon: %f E8 m \n", r_m/1E8);
printf("Orbital Radius of Moon: %f Moon Distances \n", r_m/
3.84E8);
printf("Radius of Moon: %f E6 m \n", R_m/1E6);
printf("Radius of Moon: %f Moon Radii \n", R_m/1.7374E6);
printf("Mass of Moon: %f E22 kg \n", M_m/1E22);
printf("Mass of Moon %f Moon Masses \n", M_m/7.347673E22);
printf("density of moon: %f g/cm3 \n", MoonDensity);
printf("\n");
printf("\n");
v_m=sqrt(G*M_p/r_m);
KE_m=0.5*M_m*v_m*v_m;
PDCT=(KE_m/KE_p)*(T_p);
printf("Orbital Velocity of Moon: %f m/s \n", v_m);
printf("PlanetDay Characteristic Time: %f seconds \n", PDCT);
return 0;
}}}
of 27 74
Running For Our Solar System (G2V Star System)
What is the radius of the star in solar radii? 1
What is the mass of the star in solar masses? 1
What is the luminosity of the star in solar luminosities? 1
What is the radius of the planet in Earth radii? May we suggest it is
given by 2StarRadius^2/PlanetOrbitalRadius? 1=yes, 0=no 0
What is the planet radius in Earth radii?: 1
What is the mass of the planet in Earth masses? May we suggest that it
is such that it has the density of the Earth? 1=yes, 0=no 0
What is the mass of the planet in Earth Masses? 1
The mass of the Planet is 5.972000 E24kg
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary disc?
2.5
What integer do you want to use, F0V=2,G0V=1? 1
Choose copper-silver civilization (1.7) or gold silver (1.8)? 1.8
Angular Momentum of Planet: 7.066666 E33
planet orbital velocity: 29788.980469 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 1.496000 E11 m
planet orbital radius: 1.000000 Earth distances
planet KE: 2.649727 E33 J
planet density: 5.495144 g/cm3
hbarstar: 2.826666 E33 Js
characteristic time: 1.066777 seconds
Orbital Radius of Moon: 3.868556 E8 m
Orbital Radius of Moon: 1.007436 Moon Distances
Radius of Moon: 1.800688 E6 m
Radius of Moon: 1.036427 Moon Radii
Mass of Moon: 7.207029 E22 kg
Mass of Moon 0.980859 Moon Masses
density of moon: 2.946819 g/cm3
Orbital Velocity of Moon: 1015.029358 m/s
PlanetDay Characteristic Time: 1.210587 seconds
Program ended with exit code: 0
of 28 74
Finding The Smooth Curves I have honed the program for modeling star systems, eliminating
questions pertaining to copper-silver civilizations and silver-gold civilizations (was part of
another paper). To compute the moon’s orbital radius I just use!
!
Because we know it works for our star system. I compute the radius of the planet using!
!
But, give the option of putting in your own radius. I run the program for F5V stars, through GV
stars, to K3V stars and I use this equation to compute the radii of the planets because, again,
we know it works for our star system, and further we found given the way the radius of a star
varies with with luminosity in the HR diagram, this equation always gives a planet around the
size of the Earth. I feel this size is ideal for planets with sophisticated life because of the laws of
chemistry determining a functional density for the planet having water and the right gravity. As
such I always use the planet day as one Earth day, which again I feel is optimal for life in terms
of climate. So these values all constant, we only vary star mass, size, and luminosity as they
work on the HR diagram. I also vary the pressure gradient exponent now using the average
theoretical values it has for each spectral class. The trend is that it steadily decreases on
average with mass and luminosity of the star though it can go up and down depending on the
peculiarities of the system. That is how we ran it earlier. One of the reasons is that while for a
G2V star it can range on average from p= 1.7-2.1, for our Sun, a G2V star, it is actually high, it
is 2.5. However, here we will model stars with everything constant, as we said, but the pressure
gradient will gradually decrease with spectral class, and when we do a G2V star, we won’t use
the Sun’s data, but the average value for G2V stars. We will do lot’s of models, allowing no
gaps in the data for a plot, so we can get a well defined curve. We will use the upper value for
p in instances here. The average pressure exponents by spectral class are given in the
following table…"
r
m
= R
⊙
Ag
Au
=
R
⋆
(1.8)
R
e
=
2R
2
⊙
r
e
of 29 74
Now we run the program using this for all these a spectral types…."
of 30 74
F5V Star
What is the radius of the star in solar radii? 1.473
What is the mass of the star in solar masses? 1.33
What is the luminosity of the star in solar luminosities? 3.63
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 2.4
Angular Momentum of Planet: 9.321447 E33
PlanetYear: 2.280109 years
PlanetYear: 71954776.000000 seconds
planet orbital velocity: 24888.847656 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 2.850263 E11 m
planet orbital radius: 1.905256 Earth distances
planet KE: 1.849692 E33 J
planet density: 3.627237 g/cm3
hbarstar: 3.883936 E33 Js
characteristic time: 2.099775 seconds
Orbital Radius of Moon: 5.676287 E8 m
Orbital Radius of Moon: 1.478200 Moon Distances
Radius of Moon: 2.042695 E6 m
Radius of Moon: 1.175720 Moon Radii
Mass of Moon: 7.107576 E22 kg
Mass of Moon 0.967323 Moon Masses
density of moon: 1.990782 g/cm3
Orbital Velocity of Moon: 837.955261 m/s
PlanetDay Characteristic Time: 1.165595 seconds
Program ended with exit code: 0
of 31 74
F6V Star
What is the radius of the star in solar radii? 1.359
What is the mass of the star in solar masses? 1.25
What is the luminosity of the star in solar luminosities? 2.69
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 2.4
Angular Momentum of Planet: 9.113905 E33
PlanetYear: 1.878498 years
PlanetYear: 59280888.000000 seconds
planet orbital velocity: 26005.939453 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 2.453622 E11 m
planet orbital radius: 1.640122 Earth distances
planet KE: 2.019458 E33 J
planet density: 3.751840 g/cm3
hbarstar: 3.797460 E33 Js
characteristic time: 1.880435 seconds
Orbital Radius of Moon: 5.236982 E8 m
Orbital Radius of Moon: 1.363797 Moon Distances
Radius of Moon: 2.019827 E6 m
Radius of Moon: 1.162557 Moon Radii
Mass of Moon: 7.263965 E22 kg
Mass of Moon 0.988608 Moon Masses
density of moon: 2.104477 g/cm3
Orbital Velocity of Moon: 872.393616 m/s
PlanetDay Characteristic Time: 1.182626 seconds
Program ended with exit code: 0
of 32 74
F7V
What is the radius of the star in solar radii? 1.324
What is the mass of the star in solar masses? 1.21
What is the luminosity of the star in solar luminosities? 2.45
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 2.3
Angular Momentum of Planet: 9.014981 E33
PlanetYear: 1.780055 years
PlanetYear: 56174260.000000 seconds
planet orbital velocity: 26191.283203 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 2.341610 E11 m
planet orbital radius: 1.565248 Earth distances
planet KE: 2.048346 E33 J
planet density: 3.813764 g/cm3
hbarstar: 3.919557 E33 Js
characteristic time: 1.913523 seconds
Orbital Radius of Moon: 5.102108 E8 m
Orbital Radius of Moon: 1.328674 Moon Distances
Radius of Moon: 2.008835 E6 m
Radius of Moon: 1.156231 Moon Radii
Mass of Moon: 7.375834 E22 kg
Mass of Moon 1.003833 Moon Masses
density of moon: 2.172156 g/cm3
Orbital Velocity of Moon: 883.849243 m/s
PlanetDay Characteristic Time: 1.215200 seconds
Program ended with exit code: 0
of 33 74
F8V Star
What is the radius of the star in solar radii? 1.221
What is the mass of the star in solar masses? 1.18
What is the luminosity of the star in solar luminosities? 1.95
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 2.3
Angular Momentum of Planet: 8.192318 E33
PlanetYear: 1.518926 years
PlanetYear: 47933648.000000 seconds
planet orbital velocity: 27383.435547 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 2.089050 E11 m
planet orbital radius: 1.396424 Earth distances
planet KE: 2.239060 E33 J
planet density: 4.402413 g/cm3
hbarstar: 3.561877 E33 Js
characteristic time: 1.590791 seconds
Orbital Radius of Moon: 4.705191 E8 m
Orbital Radius of Moon: 1.225310 Moon Distances
Radius of Moon: 1.914985 E6 m
Radius of Moon: 1.102213 Moon Radii
Mass of Moon: 7.359460 E22 kg
Mass of Moon 1.001604 Moon Masses
density of moon: 2.501858 g/cm3
Orbital Velocity of Moon: 920.374084 m/s
PlanetDay Characteristic Time: 1.202798 seconds
Program ended with exit code: 0
of 34 74
F9V Stars
What is the radius of the star in solar radii? 1.167
What is the mass of the star in solar masses? 1.13
What is the luminosity of the star in solar luminosities? 1.66
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 2.2
Angular Momentum of Planet: 8.030715 E33
PlanetYear: 1.375604 years
PlanetYear: 43410752.000000 seconds
planet orbital velocity: 27897.664062 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 1.927461 E11 m
planet orbital radius: 1.288410 Earth distances
planet KE: 2.323943 E33 J
planet density: 4.535965 g/cm3
hbarstar: 3.650325 E33 Js
characteristic time: 1.570746 seconds
Orbital Radius of Moon: 4.497100 E8 m
Orbital Radius of Moon: 1.171120 Moon Distances
Radius of Moon: 1.896003 E6 m
Radius of Moon: 1.091288 Moon Radii
Mass of Moon: 7.512481 E22 kg
Mass of Moon 1.022430 Moon Masses
density of moon: 2.631352 g/cm3
Orbital Velocity of Moon: 941.427246 m/s
PlanetDay Characteristic Time: 1.237699 seconds
Program ended with exit code: 0
of 35 74
G0V Star
What is the radius of the star in solar radii? 1.100
What is the mass of the star in solar masses? 1.06
What is the luminosity of the star in solar luminosities? 1.35
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 2.2
Angular Momentum of Planet: 7.794995 E33
PlanetYear: 1.216323 years
PlanetYear: 38384236.000000 seconds
planet orbital velocity: 28452.804688 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 1.738195 E11 m
planet orbital radius: 1.161895 Earth distances
planet KE: 2.417352 E33 J
planet density: 4.743263 g/cm3
hbarstar: 3.543180 E33 Js
characteristic time: 1.465727 seconds
Orbital Radius of Moon: 4.238911 E8 m
Orbital Radius of Moon: 1.103883 Moon Distances
Radius of Moon: 1.867970 E6 m
Radius of Moon: 1.075153 Moon Radii
Mass of Moon: 7.536599 E22 kg
Mass of Moon 1.025712 Moon Masses
density of moon: 2.760442 g/cm3
Orbital Velocity of Moon: 969.674255 m/s
PlanetDay Characteristic Time: 1.266400 seconds
Program ended with exit code: 0
of 36 74
G1V Star
What is the radius of the star in solar radii? 1.060
What is the mass of the star in solar masses? 1.03
What is the luminosity of the star in solar luminosities? 1.20
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 2.2
Angular Momentum of Planet: 7.561726 E33
PlanetYear: 1.129585 years
PlanetYear: 35646988.000000 seconds
planet orbital velocity: 28885.433594 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 1.638786 E11 m
planet orbital radius: 1.095445 Earth distances
planet KE: 2.491424 E33 J
planet density: 4.964432 g/cm3
hbarstar: 3.437148 E33 Js
characteristic time: 1.379592 seconds
Orbital Radius of Moon: 4.084769 E8 m
Orbital Radius of Moon: 1.063742 Moon Distances
Radius of Moon: 1.839808 E6 m
Radius of Moon: 1.058943 Moon Radii
Mass of Moon: 7.536094 E22 kg
Mass of Moon 1.025644 Moon Masses
density of moon: 2.888963 g/cm3
Orbital Velocity of Moon: 987.800598 m/s
PlanetDay Characteristic Time: 1.275032 seconds
Program ended with exit code: 0
of 37 74
G2V Star (Like Our Sun)
What is the radius of the star in solar radii? 1.012
What is the mass of the star in solar masses? 1.00
What is the luminosity of the star in solar luminosities? 1.02
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 2.1
Angular Momentum of Planet: 7.390958 E33
PlanetYear: 1.014849 years
PlanetYear: 32026214.000000 seconds
planet orbital velocity: 29641.869141 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 1.510886 E11 m
planet orbital radius: 1.009951 Earth distances
planet KE: 2.623620 E33 J
planet density: 5.137477 g/cm3
hbarstar: 3.519504 E33 Js
characteristic time: 1.341468 seconds
Orbital Radius of Moon: 3.899798 E8 m
Orbital Radius of Moon: 1.015572 Moon Distances
Radius of Moon: 1.818914 E6 m
Radius of Moon: 1.046918 Moon Radii
Mass of Moon: 7.727846 E22 kg
Mass of Moon 1.051741 Moon Masses
density of moon: 3.065734 g/cm3
Orbital Velocity of Moon: 1010.955322 m/s
PlanetDay Characteristic Time: 1.300484 seconds
Program ended with exit code: 0
of 38 74
G3V Star
What is the radius of the star in solar radii? 1.002
What is the mass of the star in solar masses? 0.99
What is the luminosity of the star in solar luminosities? 0.98
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 2.1
Angular Momentum of Planet: 7.393050 E33
PlanetYear: 0.989814 years
PlanetYear: 31236148.000000 seconds
planet orbital velocity: 29789.738281 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 1.480965 E11 m
planet orbital radius: 0.989950 Earth distances
planet KE: 2.649862 E33 J
planet density: 5.135297 g/cm3
hbarstar: 3.520500 E33 Js
characteristic time: 1.328560 seconds
Orbital Radius of Moon: 3.861263 E8 m
Orbital Radius of Moon: 1.005537 Moon Distances
Radius of Moon: 1.819172 E6 m
Radius of Moon: 1.047066 Moon Radii
Mass of Moon: 7.754257 E22 kg
Mass of Moon 1.055335 Moon Masses
density of moon: 3.074905 g/cm3
Orbital Velocity of Moon: 1015.987427 m/s
PlanetDay Characteristic Time: 1.304901 seconds
Program ended with exit code: 0
of 39 74
G4V Star
What is the radius of the star in solar radii? 0.991
What is the mass of the star in solar masses? 0.985
What is the luminosity of the star in solar luminosities? 0.91
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 2.0
Angular Momentum of Planet: 7.617845 E33
PlanetYear: 0.938673 years
PlanetYear: 29622276.000000 seconds
planet orbital velocity: 30270.066406 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 1.427093 E11 m
planet orbital radius: 0.953939 Earth distances
planet KE: 2.736003 E33 J
planet density: 4.909676 g/cm3
hbarstar: 3.808922 E33 Js
characteristic time: 1.392148 seconds
Orbital Radius of Moon: 3.818874 E8 m
Orbital Radius of Moon: 0.994498 Moon Distances
Radius of Moon: 1.846622 E6 m
Radius of Moon: 1.062865 Moon Radii
Mass of Moon: 8.045825 E22 kg
Mass of Moon 1.095017 Moon Masses
density of moon: 3.050349 g/cm3
Orbital Velocity of Moon: 1021.610596 m/s
PlanetDay Characteristic Time: 1.325893 seconds
Program ended with exit code: 0
of 40 74
G5V Star
What is the radius of the star in solar radii? 0.977
What is the mass of the star in solar masses? 0.98
What is the luminosity of the star in solar luminosities? 0.89
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 2.0
Angular Momentum of Planet: 7.358124 E33
PlanetYear: 0.925510 years
PlanetYear: 29206866.000000 seconds
planet orbital velocity: 30361.355469 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 1.411324 E11 m
planet orbital radius: 0.943398 Earth distances
planet KE: 2.752531 E33 J
planet density: 5.171903 g/cm3
hbarstar: 3.679062 E33 Js
characteristic time: 1.336611 seconds
Orbital Radius of Moon: 3.764924 E8 m
Orbital Radius of Moon: 0.980449 Moon Distances
Radius of Moon: 1.814870 E6 m
Radius of Moon: 1.044589 Moon Radii
Mass of Moon: 7.969311 E22 kg
Mass of Moon 1.084603 Moon Masses
density of moon: 3.182711 g/cm3
Orbital Velocity of Moon: 1028.904175 m/s
PlanetDay Characteristic Time: 1.324104 seconds
Program ended with exit code: 0
of 41 74
G6V
What is the radius of the star in solar radii? 0.949
What is the mass of the star in solar masses? 0.97
What is the luminosity of the star in solar luminosities? 0.79
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 2.0
Angular Momentum of Planet: 7.379326 E33
PlanetYear: 0.850719 years
PlanetYear: 26846640.000000 seconds
planet orbital velocity: 31119.650391 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 1.329674 E11 m
planet orbital radius: 0.888819 Earth distances
planet KE: 2.891740 E33 J
planet density: 5.149630 g/cm3
hbarstar: 3.689663 E33 Js
characteristic time: 1.275932 seconds
Orbital Radius of Moon: 3.657024 E8 m
Orbital Radius of Moon: 0.952350 Moon Distances
Radius of Moon: 1.817483 E6 m
Radius of Moon: 1.046093 Moon Radii
Mass of Moon: 8.109231 E22 kg
Mass of Moon 1.103646 Moon Masses
density of moon: 3.224643 g/cm3
Orbital Velocity of Moon: 1043.972534 m/s
PlanetDay Characteristic Time: 1.320330 seconds
Program ended with exit code: 0
of 42 74
G7V Star
What is the radius of the star in solar radii? 0.927
What is the mass of the star in solar masses? 0.95
What is the luminosity of the star in solar luminosities? 0.74
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 1.9
Angular Momentum of Planet: 7.172427 E33
PlanetYear: 0.818490 years
PlanetYear: 25829588.000000 seconds
planet orbital velocity: 31304.695312 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 1.286908 E11 m
planet orbital radius: 0.860233 Earth distances
planet KE: 2.926232 E33 J
planet density: 5.374052 g/cm3
hbarstar: 3.774961 E33 Js
characteristic time: 1.290042 seconds
Orbital Radius of Moon: 3.572246 E8 m
Orbital Radius of Moon: 0.930272 Moon Distances
Radius of Moon: 1.791823 E6 m
Radius of Moon: 1.031324 Moon Radii
Mass of Moon: 8.203606 E22 kg
Mass of Moon 1.116490 Moon Masses
density of moon: 3.404338 g/cm3
Orbital Velocity of Moon: 1056.287964 m/s
PlanetDay Characteristic Time: 1.351277 seconds
Program ended with exit code: 0
of 43 74
G8V Star
What is the radius of the star in solar radii? 0.914
What is the mass of the star in solar masses? 0.94
What is the luminosity of the star in solar luminosities? 0.68
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 1.8
Angular Momentum of Planet: 7.376574 E33
PlanetYear: 0.772270 years
PlanetYear: 24370994.000000 seconds
planet orbital velocity: 31804.775391 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 1.233633 E11 m
planet orbital radius: 0.824621 Earth distances
planet KE: 3.020470 E33 J
planet density: 5.152511 g/cm3
hbarstar: 4.098097 E33 Js
characteristic time: 1.356775 seconds
Orbital Radius of Moon: 3.522150 E8 m
Orbital Radius of Moon: 0.917226 Moon Distances
Radius of Moon: 1.817144 E6 m
Radius of Moon: 1.045898 Moon Radii
Mass of Moon: 8.520856 E22 kg
Mass of Moon 1.159667 Moon Masses
density of moon: 3.390224 g/cm3
Orbital Velocity of Moon: 1063.773315 m/s
PlanetDay Characteristic Time: 1.379084 seconds
Program ended with exit code: 0
of 44 74
G9V Star
What is the radius of the star in solar radii? 0.853
What is the mass of the star in solar masses? 0.90
What is the luminosity of the star in solar luminosities? 0.55
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 1.8
Angular Momentum of Planet: 6.918504 E33
PlanetYear: 0.673135 years
PlanetYear: 21242532.000000 seconds
planet orbital velocity: 32816.039062 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 1.109463 E11 m
planet orbital radius: 0.741620 Earth distances
planet KE: 3.215601 E33 J
planet density: 5.672607 g/cm3
hbarstar: 3.843613 E33 Js
characteristic time: 1.195302 seconds
Orbital Radius of Moon: 3.287083 E8 m
Orbital Radius of Moon: 0.856011 Moon Distances
Radius of Moon: 1.759819 E6 m
Radius of Moon: 1.012904 Moon Radii
Mass of Moon: 8.516574 E22 kg
Mass of Moon 1.159085 Moon Masses
density of moon: 3.730558 g/cm3
Orbital Velocity of Moon: 1101.153198 m/s
PlanetDay Characteristic Time: 1.387337 seconds
Program ended with exit code: 0
of 45 74
K0V Star
What is the radius of the star in solar radii? 0.813
What is the mass of the star in solar masses? 0.88
What is the luminosity of the star in solar luminosities? 0.46
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 1.7
Angular Momentum of Planet: 6.826268 E33
PlanetYear: 0.595358 years
PlanetYear: 18788086.000000 seconds
planet orbital velocity: 33931.843750 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 1.014637 E11 m
planet orbital radius: 0.678233 Earth distances
planet KE: 3.437991 E33 J
planet density: 5.787967 g/cm3
hbarstar: 4.015451 E33 Js
characteristic time: 1.167964 seconds
Orbital Radius of Moon: 3.132941 E8 m
Orbital Radius of Moon: 0.815870 Moon Distances
Radius of Moon: 1.748049 E6 m
Radius of Moon: 1.006129 Moon Radii
Mass of Moon: 8.836441 E22 kg
Mass of Moon 1.202618 Moon Masses
density of moon: 3.949386 g/cm3
Orbital Velocity of Moon: 1127.916382 m/s
PlanetDay Characteristic Time: 1.412571 seconds
Program ended with exit code: 0
of 46 74
K1V Star
What is the radius of the star in solar radii? 0.797
What is the mass of the star in solar masses? 0.86
What is the luminosity of the star in solar luminosities? 0.41
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 1.7
Angular Momentum of Planet: 7.073402 E33
PlanetYear: 0.552446 years
PlanetYear: 17433884.000000 seconds
planet orbital velocity: 34523.027344 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 0.957907 E11 m
planet orbital radius: 0.640312 Earth distances
planet KE: 3.558833 E33 J
planet density: 5.487297 g/cm3
hbarstar: 4.160825 E33 Js
characteristic time: 1.169154 seconds
Orbital Radius of Moon: 3.071284 E8 m
Orbital Radius of Moon: 0.799813 Moon Distances
Radius of Moon: 1.779410 E6 m
Radius of Moon: 1.024180 Moon Radii
Mass of Moon: 9.045376 E22 kg
Mass of Moon 1.231053 Moon Masses
density of moon: 3.832757 g/cm3
Orbital Velocity of Moon: 1139.181885 m/s
PlanetDay Characteristic Time: 1.424915 seconds
Program ended with exit code: 0
of 47 74
K2V Star
What is the radius of the star in solar radii? 0.783
What is the mass of the star in solar masses? 0.82
What is the luminosity of the star in solar luminosities? 0.37
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 1.6
Angular Momentum of Planet: 7.301705 E33
PlanetYear: 0.523837 years
PlanetYear: 16531028.000000 seconds
planet orbital velocity: 34586.937500 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 0.909981 E11 m
planet orbital radius: 0.608276 Earth distances
planet KE: 3.572021 E33 J
planet density: 5.231962 g/cm3
hbarstar: 4.563565 E33 Js
characteristic time: 1.277586 seconds
Orbital Radius of Moon: 3.017334 E8 m
Orbital Radius of Moon: 0.785764 Moon Distances
Radius of Moon: 1.807899 E6 m
Radius of Moon: 1.040577 Moon Radii
Mass of Moon: 9.339789 E22 kg
Mass of Moon 1.271122 Moon Masses
density of moon: 3.773355 g/cm3
Orbital Velocity of Moon: 1149.320923 m/s
PlanetDay Characteristic Time: 1.492071 seconds
Program ended with exit code: 0
of 48 74
K3V Star
What is the radius of the star in solar radii? 0.755
What is the mass of the star in solar masses? 0.78
What is the luminosity of the star in solar luminosities? 0.28
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary
disc? 1.5
Angular Momentum of Planet: 8.340819 E33
PlanetYear: 0.435785 years
PlanetYear: 13752343.000000 seconds
planet orbital velocity: 36167.078125 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet orbital radius: 0.791609 E11 m
planet orbital radius: 0.529150 Earth distances
planet KE: 3.905860 E33 J
planet density: 4.285367 g/cm3
hbarstar: 5.560546 E33 Js
characteristic time: 1.423642 seconds
Orbital Radius of Moon: 2.909435 E8 m
Orbital Radius of Moon: 0.757665 Moon Distances
Radius of Moon: 1.932263 E6 m
Radius of Moon: 1.112158 Moon Radii
Mass of Moon: 10.277222 E22 kg
Mass of Moon 1.398704 Moon Masses
density of moon: 3.400867 g/cm3
Orbital Velocity of Moon: 1170.438843 m/s
PlanetDay Characteristic Time: 1.557185 seconds
Program ended with exit code: 0
of 49 74
Here is a plot of these results
We see the characteristic time decreases as a curve to intersect
close to a characteristic time of one second with a planet day
characteristic time of one second as a straight line at a G3V
star with G1V to G5V stars doing close to the same, which is the
region where our Sun is and may have something to do with it
being so optimal for life. We note here that this uses the
average value for G2V star, and our Sun comes closer to a second
because its pressure exponent is higher than on average, it is
very high, or steep (p=2.5) which means the pressure of its disc
drops rapidly with distance.
Planet Day Characteristic Time:
Characteristic time: ,
I have now gone ahead and included in the program for it to output the lunar orbital period in
seconds and days. Here is the program to date…"
1secon d =
K E
m
K E
e
(Ear th Da y)
ℏ
⊙
= (1secon d )KE
e
ℏ
2
⊙
GM
3
m
⋅
1
c
= 1secon d
of 50 74
//
// main.c
// Intersections
//
// Created by Ian Beardsley on 1/7/25.
//
#include <stdio.h>
#include <math.h>
int main(int argc, const char * argv[]) {
float R_p, M_p, R_s, M_s, L_s, h_s, t_c, M_m, rho_m, rho_p,
PlanetDay, V_p,StarRadius, PlanetRadius, PlanetMass, StarLuminosity,
PlanetOrbit, StarMass, r_p, T_p, p, L_p, KE_p, v_p, Root, T_m,Tmoon,
C_m;
float G=6.674E-11, hbarstar, PDCT,Tsquared,T,PlanetYear;
float Mcubed, r_m, root, R_m, V_m, MoonDensity, part1, part2,
part3,v_m, KE_m;
int i;
printf ("What is the radius of the star in solar radii? ");
scanf ("%f", &StarRadius);
printf ("What is the mass of the star in solar masses? ");
scanf ("%f", &StarMass);
printf ("What is the luminosity of the star in solar luminosities?
");
scanf ("%f", &StarLuminosity);
PlanetOrbit=sqrt(StarLuminosity);
r_p=PlanetOrbit*1.496E11;
M_s=1.9891E30*StarMass;
Tsquared=((4*3.14159*3.14159)/(G*M_s))*r_p*r_p*r_p;
T=sqrt(Tsquared);
PlanetYear=T/31557600;
printf("Do you want us to compute the planet radius, 1=yes, 0=no?
");
scanf("%i", &i);
R_s=6.9364E8*StarRadius;
if (i==1)
{
R_s=6.9364E8*StarRadius;
R_p=2*(R_s*R_s)/r_p;
PlanetRadius=R_p/6.378E6;
}
else
{
printf("What is the planet radius in Earth radii?: ");
scanf("%f", &PlanetRadius);
of 51 74
R_p=PlanetRadius*6.378E6;
}
printf("What is the mass of the planet in Earth
masses? ");
scanf("%f", &PlanetMass);
M_p=PlanetMass*5.972E24;
printf ("What is the planet day in Earth days? ");
scanf ("%f", &PlanetDay);
T_p=PlanetDay*86400;
printf("That is %f seconds \n", T_p);
{
printf("What is p the pressure gradient exponent
of the protoplanetary disc? ");
scanf("%f", &p);
M_s=1.9891E30*StarMass;
r_m=R_s/1.8;
v_p=sqrt(G*M_s/r_p);
L_p=0.8*3.14159*M_p*(1/T_p)*R_p*R_p;
KE_p=0.5*M_p*v_p*v_p;
hbarstar=L_p/p;
t_c=hbarstar/KE_p;
part1=cbrt(hbarstar/(t_c));
part2=cbrt(1/G);
part3=cbrt(hbarstar/299792458);
M_m=part1*part2*part3;
R_s=StarRadius*6.9634E8;
R_m=R_s*r_m/r_p;
V_m=1.33333*3.14159*R_m*R_m*R_m;
rho_m=(M_m/V_m);
MoonDensity=rho_m*0.001;
V_p=1.33333*3.14159*R_p*R_p*R_p;
rho_p=(M_p/V_p)*0.001;
printf("\n");
printf("\n");
of 52 74
printf("Angular Momentum of Planet: %f E33 \n",
L_p/1E33);
printf("\n");
printf("\n");
printf("PlanetYear: %f years \n", PlanetYear);
printf("PlanetYear: %f seconds \n", T);
printf("planet orbital velocity: %f m/s \n", v_p);
printf("planet mass: %f E24 kg \n", M_p/1E24);
printf("planet mass: %f Earth masses \n", M_p/
5.972E24);
printf("planet radius %f meters \n", R_p);
printf("planet radius: %f Earth Radii \n",
PlanetRadius);
printf("planet orbital radius: %f E11 m \n", r_p/
1E11);
printf ("planet orbital radius: %f Earth distances
\n", r_p/1.496E11);
printf("planet KE: %f E33 J \n",KE_p/1E33);
printf("planet density: %f g/cm3 \n", rho_p);
printf("\n");
printf("\n");
printf("hbarstar: %f E33 Js \n", hbarstar/1E33);
printf("characteristic time: %f seconds\n", t_c);
printf("\n");
printf("\n");
printf("Orbital Radius of Moon: %f E8 m \n", r_m/
1E8);
printf("Orbital Radius of Moon: %f Moon Distances
\n", r_m/3.84E8);
printf("Radius of Moon: %f E6 m \n", R_m/1E6);
printf("Radius of Moon: %f Moon Radii \n", R_m/
1.7374E6);
printf("Mass of Moon: %f E22 kg \n", M_m/1E22);
printf("Mass of Moon %f Moon Masses \n", M_m/
7.347673E22);
printf("density of moon: %f g/cm3 \n",
MoonDensity);
printf("\n");
printf("\n");
v_m=sqrt(G*M_p/r_m);
KE_m=0.5*M_m*v_m*v_m;
PDCT=(KE_m/KE_p)*(T_p);
printf("Orbital Velocity of Moon: %f m/s \n",
v_m);
printf("PlanetDay Characteristic Time: %f seconds
\n", PDCT);
C_m=2*3.14159*r_m;
of 53 74
T_m=C_m/v_m;
Tmoon=T_m*(1.0/24)*(1.0/60)*(1.0/60);
printf("Lunar Orbital Period: %f seconds \n",
T_m);
printf("Lunar Orbital Period: %f days \n", Tmoon);
return 0;
}}
of 54 74
What is the radius of the star in solar radii? 1
What is the mass of the star in solar masses? 1
What is the luminosity of the star in solar luminosities? 1
Do you want us to compute the planet radius, 1=yes, 0=no? 1
What is the mass of the planet in Earth masses? 1
What is the planet day in Earth days? 1
That is 86400.000000 seconds
What is p the pressure gradient exponent of the protoplanetary disc?
2.5
Angular Momentum of Planet: 7.187518 E33
PlanetYear: 0.999888 years
PlanetYear: 31554074.000000 seconds
planet orbital velocity: 29788.980469 m/s
planet mass: 5.972000 E24 kg
planet mass: 1.000000 Earth masses
planet radius 6432306.000000 meters
planet radius: 1.008515 Earth Radii
planet orbital radius: 1.496000 E11 m
planet orbital radius: 1.000000 Earth distances
planet KE: 2.649727 E33 J
planet density: 5.357135 g/cm3
hbarstar: 2.875007 E33 Js
characteristic time: 1.085020 seconds
Orbital Radius of Moon: 3.853556 E8 m
Orbital Radius of Moon: 1.003530 Moon Distances
Radius of Moon: 1.793707 E6 m
Radius of Moon: 1.032408 Moon Radii
Mass of Moon: 7.247882 E22 kg
Mass of Moon 0.986419 Moon Masses
density of moon: 2.998263 g/cm3
Orbital Velocity of Moon: 1017.002930 m/s
PlanetDay Characteristic Time: 1.222188 seconds
Lunar Orbital Period: 2380778.000000 seconds
Lunar Orbital Period: 27.555302 days
Program ended with exit code: 0
of 55 74
Here we fit the curves for characteristic time and planet day characteristic time. We name the
spectral types with number for input according to the following scheme.!
F5V is 1.5, F6V is 1.6, F7V is 1.7,…G0V is 2.0, G1V is 2.1,…!
For the characteristic time we fit the curve with a power law decay!
!
For the planet day characteristic time we fit the curve with a straight line!
!
Where we have chosen in ,!
!
!
We intend to fit the curves in!
y = 2.8x
−
3
2
x
+ 1.1
y = 0.168x + 0.913595
y = m x + b
m =
G3V − F 9V
2.3 − 1.9
= 0.168
b = 1.165595 − 0.168(1.5)
of 56 74
The results are"
of 57 74
!
of 58 74
A Theory For Biological Hydrocarbons I have found that the basis unit of one second is
not just a Natural constant for physical systems like the atom and the planets around the Sun
but for the basis of biological life, that it is in the sixfold Nature of the chemical skeletons from
which life is built, the hydrocarbons. I found
1).
2).
Where is the fine structure constant, is the radius of a proton, is the mass of a
proton, is Planck’s constant, is the constant of gravitation, is the speed of light, is the
kinetic energy of the Moon, is the kinetic energy of the Earth, and is the
rotation period of the Earth is one day.
The first can be written:
3).
4).
From which instead of saying the left sides of these equations are seconds, we say they are
proton-seconds by not letting the units of cancel with the bodies of these equations on the
left, which are in units of mass, but rather divide into them, giving us a number of protons. I say
this is the biological because these are the hydrocarbons the backbones of biological chemistry.
We see they display sixfold symmetry. I can generate integer numbers of protons from the time
values from these equations for all of the elements with a computer program. Some results are:
1secon d =
1
6α
2
⋅
r
p
m
p
4πh
Gc
1secon d =
K E
m
K E
e
(Ear th Day)
α = 1/137
r
p
m
p
h
G
c
K E
m
K E
e
Ear th Day
1
α
2
m
p
h 4π r
2
p
Gc
= 6proton ⋅ secon ds = carbon(C )
1
α
2
m
p
h 4π r
2
p
Gc
= 1proton ⋅ 6secon ds = hydrogen(H )
m
p
of 59 74
A very interesting thing here is looking at the values generated by the program, the smallest
integer value 1 second produces 6 protons (carbon) and the largest integer value 6 seconds
produces one proton (hydrogen). Beyond six seconds you have fractional protons, and the rest of
the elements heavier than carbon are formed by fractional seconds. These are the hydrocarbons
the backbones of biological chemistry. And carbon is the core element of life. We see the
duration of the base unit of measuring time, 1 second, given to us by the ancients (the base 60,
sexagesimal, system of counting of the Sumerians who invented math and writing and started
civilization), is perfect for the mathematical formulation of life chemistry. Here is the code for
the program, it finds integer solutions for time values, incremented by the program at the
discretion of the user:
#include <stdio.h>
#include <math.h>
int main(int argc, const char * argv[]) {
int n;
float value=0, increment,t=0, p=1.67262E-27, h=6.62607E-34,G=6.67408E-11,
c=299792458,protons[100],r=0.833E-15;
do
{
printf("By what value would you like to increment?: ");
scanf("%f", &increment);
printf("How many values would you like to calculate for t in equation 1 (no more than 100?): ");
scanf("%i", &n);
}
while (n>=101);
{
for (int i=0; i<n;i++)
{
protons[i]=((137*137)/(t*p))*sqrt(h*4*(3.14159)*(r*r)/(G*c));
int intpart=(int)protons[i];
float decpart=protons[i]-intpart;
t=t+increment;
if (decpart<0.25)
{ printf("%.4f protons %f seconds %f decpart \n", protons[i], t-increment, decpart);
}}}}
We have that
Since this 6 seconds is also proton-seconds we have
5). is carbon (C)
6). is hydrogen (H)
1
α
2
⋅
r
p
m
p
4πh
G c
= 18769
0.833E − 15
1.67262E − 27
4π (6.62607E − 34)
(6.67408E − 11)(299,792458)
= 6.029978s ≈ 6s
1
6pr oton s
⋅
1
α
2
⋅
r
p
m
p
4πh
G c
= 1secon d
1
1pr oton
⋅
1
α
2
⋅
r
p
m
p
4πh
G c
= 6secon d s
of 60 74
Our Theory For Hydrocarbons With all that has been said we are equipped to proceed. We
want to consider the radius of a hydrogen atom and the radius of a carbon atom. The radius of
a carbon atom given in your periodic table of the elements is often 70 to 76 picometers. The
covalent radius of hydrogen is given as 31 picometers. The atomic radius of hydrogen is 53
picometers and the atomic radius of carbon is 67 picometers. We want to consider the atomic
radii of both, because the covalent radius, determined by x-ray diffraction for diatomic
hydrogen, is the size of two hydrogen atoms joined H2 divided by two, where it is measured
that way, joined, in the laboratory. Carbon is C2 divided by 2. We are interested in the single
carbon and hydrogen atoms, because we want to know what our theory for their six-fold
symmetry with one another in their representations in proton-seconds says about the way they
combine as the skeletons of life chemistry. We start with the Planck constant, which is like
flux, a mass (perhaps a number of particles) per second over an area. That is it is kilograms per
second over square area:!
!
We have equations 1 and 2:!
1). is carbon (C)
2). is hydrogen (H)
We can write these
=
3).
=
4). !
We have from 3 and 4…!
5). !
We find the ratio between the surface areas of the hydrogen and carbon atoms:!
h,
h = 6.62607E − 34J ⋅ s = 6.62607E − 37
kg
s
⋅ m
2
1
6pr oton s
⋅
1
α
2
⋅
r
p
m
p
4πh
G c
= 1secon d
1
1pr oton
⋅
1
α
2
⋅
r
p
m
p
4πh
G c
= 6secon d s
(
6.62607E − 37
kg
s
⋅ m
2
)
6secon d s
m
p
(
6.62607E − 37
kg
s
⋅ m
2
)
6seconds
1.67262E − 27kg
= 2.37689E − 6m
2
(
6.62607E − 37
kg
s
⋅ m
2
)
1second
6m
p
(
6.62607E − 37
kg
s
⋅ m
2
)
1second
6(1.67262E − 27kg)
= 6.602486E − 8m
2
h
m
p
⋅
(6seconds)
1
h
m
p
⋅
(1second )
6
=
2.37689E − 6m
2
6.602486E − 8m
2
= 35.9999 ≈ 40
of 61 74
6). !
7). !
!
It is the golden mean. These atomic radii are the radii between the nucleus of the atoms and
their valence shell, which is what we want because the valence shell is the outermost electrons
responsible for the way the hydrogen and the carbon combine to make hydrocarbons. We will
write this!
8). !
I am guessing the reason we have the golden mean here is that it is the number used for
closest packing. But what we really want to do is look at the concept of action, for hydrogen
given by six seconds and carbon given by 1 second. We take equations 1 and 2:!
1). is carbon (C)
2). is hydrogen (H)
And, we write
9).
Where is the radius of the atom, and t its time values given here by equations 1 and 2. We have for
hydrogen
10).
=
This is actually very close to the radius of a hydrogen which can vary around this depending on how you
are looking at it, which we said is given by 5.3E-11m. For carbon we have:
11).
=
And this is actually very close to the radius of a carbon atom which is 6.7E-11m. The thing is if we consider
the bond length of the simplest hydrocarbon CH4, methane, which can be thought of as
H
surf ace
= 4π (r
2
H
) = 3.52989E − 20m
2
C
surf ace
= 4π (r
2
C
) = 5.64104E − 20m
2
4π(53pm)
2
4π(67pm)
2
= 0.62575 ≈ 0.618... = ϕ
r
2
H
r
2
C
= ϕ
HC
1
6pr oton s
⋅
1
α
2
⋅
r
p
m
p
4πh
G c
= 1secon d
1
1pr oton
⋅
1
α
2
⋅
r
p
m
p
4πh
G c
= 6secon d s
m
p
G c
h
∫
t
0
dt = r
A
r
A
m
p
G c
h
∫
t
0
dt = (1.67262E − 27kg)
(6.67408E − 11)(299,792,458)
6.62607E − 34
∫
6sec
0
dt
(9.2E − 12m /s)(6seco n d s) = 5.5E − 11m
m
p
G c
h
∫
t
0
dt = (1.67262E − 27kg)
(6.67408E − 11)(299,792,458)
6.62607E − 34
∫
6sec+1sec
0
dt
(9.2E − 12m /s)(7secon d s) = 6.44E − 11m
of 62 74
16).
our, equations give
17).
which are the same thing. Thus we have the basis for a theory of everything in that it includes the macro
scale, the Earth/Moon/Sun System, because we had:
and it includes the radius of the proton which is the radius and mass of a proton and gives 1
second
and we have the hydrocarbons the skeleton of the chemistry of life give one second
1). is carbon (C)
2). is hydrogen (H)
and because they predict the radii of the carbon and hydrogen atoms at the core of life
10).
=
11).
=
r
H
+ r
C
= 5.3E − 11m + 6.7E − 11m = 1.2E − 11m
r
H
+ r
C
= 5.5E − 11m + 6.44E − 11m = 1.2E − 11m
1secon d =
K E
m
K E
e
(Ear th Day)
1secon d =
1
6α
2
⋅
r
p
m
p
4πh
Gc
1
6pr oton s
⋅
1
α
2
⋅
r
p
m
p
4πh
G c
= 1secon d
1
1pr oton
⋅
1
α
2
⋅
r
p
m
p
4πh
G c
= 6secon d s
m
p
G c
h
∫
t
0
dt = (1.67262E − 27kg)
(6.67408E − 11)(299,792,458)
6.62607E − 34
∫
6sec
0
dt
(9.2E − 12m /s)(6seco n d s) = 5.5E − 11m
m
p
G c
h
∫
t
0
dt = (1.67262E − 27kg)
(6.67408E − 11)(299,792,458)
6.62607E − 34
∫
6sec+1sec
0
dt
(9.2E − 12m /s)(7secon d s) = 6.44E − 11m
of 63 74
Lunar Eclipses Since we have said it may be the optimization of life occurs under the
condition where the moon of a planet perfectly eclipses the star that the planet in the habitable
zone orbits as seen from from the planet, we will look at that condition a bit, which is equation
8 in the introduction!
8.
To look at this we only need to know two things: The Earth loses energy to the Moon such that
the Moon’s orbit grows (recedes) by 0.0000382 kilometers per year, and the Earth day becomes
longer because of this by 0.00002 seconds per year. Before we look at this we want to consider
equation 1.10
1.10
Which is really more accurately represented if we consider the average orbital velocity of the
Moon and the average orbital velocity of the Earth:
Where the EarthDay is 86,400 seconds. The sidereal day is a little less than this, but by so little
there is no point in using it. A variation on 1.10 is 2.2, is
2.2.
We can write this
Or,…
We want it like this to see the base 60 of the ancient Sumerians interacting with their 24 for the
hours in a day. But, we will remember we estimated 1.252 seconds to be 1 second. For the
equation to work perfectly we want to write
r
planet
r
moon
=
R
star
R
moon
1secon d =
K E
m
K E
e
(Ear th Day)
K E
m
=
1
2
(7.3476731kg)(1,022 m /s)
2
= 3.837E 28J
K E
e
=
1
2
(5.972E 24kg)(29,780m /s)
2
= 2.648E 33J
K E
m
K E
e
(Ear th Day) =
3.837E 28J
2.648E 33J
(86,400s) = 1.252secon ds
(
M
m
R
m
(Ear th Day)
)
⋅
R
⊙
M
⊙
= 1secon d
R
⊙
M
⊙
=
R
m
M
m
⋅
1secon d
Ear th Day
R
⊙
M
⊙
=
R
m
M
m
⋅
1secon d
86400sec
=
R
m
M
m
⋅
1
(24)(60)(60)
of 64 74
The Appearance of Photosynthesizers And An Oxygen Rich Atmosphere
We ask: How many years ago was the Earth day 18.794 hours long? Convert it to seconds
The Earth’s rotation slow by 0.00002 seconds per year. We have
The Earth day, then, was 18.794 hours long about 937 million years ago. Thus, how much closer
was the Moon to the Earth at this time? It was
Putting it at
It’s angular size in the sky was then
Thus, the Moon appeared to be about 0.57 degrees wide in the Earth sky. How big did the Sun
appear in the Earth sky?
So, 937 million years ago, when the Earth day was about 18 hours long, the Moon was larger in
the sky than the Sun, on the average. By how much?
The Sun was then about 94% the size of the Moon in the sky back then. At his time we could
write
R
⊙
M
⊙
=
R
m
M
m
⋅
1
(18.794)(60)(60)
(18.794hr)(60mi n)(60sec) = 67,658.4secon d s
24hrs − 18.794hrs = 5.206hrs
(5.206hrs)(60min)(60min) = 18,741.6secon ds
(18,741.6s)(yr /0.00002sec) = 937,080,000years
(937,080,000yrs)(0.0000382k m /yr) = 35,796.456k m
384,400k m − 35,796.456k m = 348,603.544 k m
θ
moon
= 2ar csin
(
R
m
r
m
)
= 2ar csin
(
1,737.4k m
348,603.544k m
)
= 0.57
∘
θ
sun
= 2ar csin
(
R
⊙
r
e
)
= 2ar csin
(
696,340k m
149,597,870.7k m
)
= 0.5334
∘
0.57
0.5334
1.0686 ≈ 107 %
(Ear th Day)
R
⊙
M
⊙
=
R
m
M
m
(1secon d )
of 65 74
We ask what era of the Earth history was this, and what was characteristic of it? It was the
Tonian Period of the Neoproterozoic Era in the Proterozoic Eon. It was the time of early steps
towards multicellular life with the appearance of multicellular eukaryotes such as red algae. It
was the beginning of abundance in photosynthesizers appearing and converting the primordial
CO2 atmosphere into an atmosphere of oxygen setting the stage for the development of
organisms that breathed oxygen to create their energy from food. It was the beginning of
multicellular life and a rich oxygen atmosphere on Earth, setting the scene for the diversification
and development of more complex life. We see 937 million years ago the Sun was about 94% the
size of the Moon in the Earth sky and we ask when was it the same size as the Moon in the Earth
sky. That is when were perfect eclipses most dominant. Today, the annual diameter of the Moon
is
The moon has an angular diameter on the average throughout a year of about a half a degree.
Let’s see what the size of the Sun is…
Thus the Moon is 97%
Now, the Moon on the average is 97% the size of the Sun throughout the year meaning annular
eclipses dominate.
The Time of Perfect Eclipses
Thus, to find when the Moon was the same size as the Sun, when there were perfect eclipses
dominating, we write
This is how far the Moon was from the Earth at the time perfect eclipses dominated. We ask how
far it has moved since then?
We ask how long did it take the Moon to move this distance. We have
θ
moon
= 2ar csin
(
R
m
r
m
)
= 2ar csin
(
1,737.4k m
384,400k m
)
= 0.518
∘
θ
sun
= 2ar csin
(
R
⊙
r
e
)
= 2ar csin
(
696,340k m
149,597,870.7k m
)
= 0.5334
∘
θ
moon
θ
sun
=
0.518
0.5334
= 97 %
0.5334
∘
2
= ar csin
(
1,737.4k m
x
)
x = 373,251k m
384,400k m − 373,251k m = 11,149k m
11,149k m = (0.0000382k m /yr)t
t = 291,858,638.7years
of 66 74
About 292 million years ago we had perfect eclipses dominating. We ask what was the Earth day
at this time? It was
At the time of perfect eclipses of the Sun by the Moon as seen from the Earth dominating the
Earth sky, which was about 292 million years ago, the Earth day was 22.38 hours long. We ask
what era was this in the Earth history and what were its characteristics. This was the Permian
Period which was from 298.9 million years to 251.9 million years ago. The Earth’s continents
were mostly joined into a single massive supercontinent called Pangea, which was surrounded
by a global ocean called Panthalassa. Ecosystems were dominated by lycophytes, ferns, and seed
ferns in wetter regions that were gradually replaced by gymnosperms, including conifers and
ginkgoes which could tolerate the drier conditions. This was a time of the the diversification of
reptiles and large amphibians continued to thrive in wetter areas. The oceans were teeming with
life such as brachiopods, bivalves, and ammonoids. The late Permian period ended around 252
million years ago in the most severe extinction event in the Earth’s history where it lost about
90% of marine species and 70% of terrestrial species, but this period set the stage for the rise of
dinosaurs in the Mesozoic Era.
When The PlanetDay Characteristic Time Was Exactly One Second
Let us ask when was the Earth day characteristic time exactly one second. That is
Which is the PlanetDay characteristic time today using average orbital velocities and an
EarthDay of 86,400 seconds. The sidereal day is a little shorter, but we don’t use it here because
the difference is too small to be noted. To find when the Earth day is such that the PlanetDay
characteristic time is exactly one second for the the Earth, we write
The Earth day had to be 69,012 seconds long compared to the 86,400 seconds it is today. That is
it was 69,012/86,400~80% what it is today. We write
(0.00002s /yr)(291,858,638.7yr) = 5837sec = 1.62hrs
24hrs − 1.62hrs = 22.38hrs
K E
m
K E
e
(Ear th Day) = 1secon d
K E
m
=
1
2
(7.3476731E 22kg)(1,022m /s)
2
= 3.837E 28J
K E
e
=
1
2
(5.972E 24kg)(29,780m /s)
2
= 2.648E 33J
K E
m
K E
e
(Ear th Day) =
3.837E 28J
2.648E 33J
(86,400s) = 1.252secon ds
K E
m
K E
e
t = 1.00seconds
t = 69,012seconds
69,012sec = 19.17hrs
of 67 74
The PlanetDay characteristic time for the Earth was 1 second around about 870 million years
ago. You might think the problem is when we go back in time to about 870 million years, the
Moon is closer to the Earth so its kinetic energy has changed due to the increase in its orbital
velocity and so the PlanetDay characteristic time is then no longer the same. Let’s look at this…
The lunar orbital radius is now
We find the orbital velocity of the Moon was
Since the average orbital velocity of the Moon today is 1,022 m/s, we see it has increased by only
0.69 m/s, which is practically nothing. So we don’t have to solve a differential equation and the
answer is then, 869,400,000 years ago the PlanetDay characteristic time for the Earth was 1
second which it remained close to for a long time, perhaps hundreds of millions of years before
and after that time.
This intersection of carbon, the proton, and planet day characteristic time and star system
characteristic time of one second, was when plants on Earth were converting the atmosphere to
oxygen leading to the more complex organisms that exist today using oxygen to burn food for
energy. Oxygen started entering the atmosphere during the great oxidation event about 2.4
billion years ago, but the turning point where oxygen began to rise more reliably and
consistently was about 900 million years ago. This is about when animals started using oxygen
for respiration. This time also marks the beginning of multicellular life.
When We Will Lose Perfect Eclipses
We ask, when will the Earth lose perfect eclipses of the Sun by the Moon. The Moon is closest to
the Earth at perihelion, which is 360,000 km. We have
So, on the average the Moon is smaller than the Sun (97%) and because the orbit is elliptical at
its closest approach it is 103.67% larger than the Sun. So we live in a time when we have annular
eclipses ( ) perfect eclipses ( ) and total eclipses ( ). But when
the closest approach of the Moon no longer has there will no longer be perfect or
86,400sec − 69,012sec = 17,388second s
(17,388sec)(yr /0.00002sec) = 869,400,000years
(869,400,000years)(0.0000382k m /year) = 3,321k m
r
m
= 384,400k m − 3,321k m = 381,079k m
v
m
=
GM
e
r
m
=
(6.674E − 11)(5.972E 24kg)
381,079,000m
= 1,022.69m /s
θ
moon
= 2ar csin
(
R
m
r
m
)
= 2ar csin
(
1,737.4k m
360,000k m
)
= 0.553
∘
θ
moon
θ
sun
=
0.553
0.5334
= 103.67454 %
θ
moon
< θ
sun
θ
moon
= θ
sun
θ
moon
> θ
sun
θ
moon
> θ
sun
of 68 74
total eclipses, only annular eclipses. Let’s find in how long that will be…A perfect eclipse we
found occurs when the Moon is at 373,251km. We have
In about 347 million years we will no longer have perfect eclipses. This is a long time from now,
perhaps longer after the Earth can reasonably sustain life and we have moved on to other
realms.
373,251k m − 360,000k m = 13,251k m
(13,251k m)(yr /0.0000382k m) = 346,884,817years
of 69 74
The Archaeological Component We can derive the condition such that all of the characteristic times
intersect including the characteristic time of the proton. Let’s do that…!
We have for the Sun!
, !
, so we write!
!
If the PlanetDay Characteristic time is one second. We also have!
, , !
We know!
, !
So we have!
, !
This becomes!
Which we can write after squaring both sides!
We put that in our first equation: !
And we have!
Condition for all three characteristics times to be 1 second.!
We write it!
!
ℏ
2
⊙
GM
3
m
⋅
1
c
= 1secon d
K E
m
K E
e
(Ear th Da y = 1.2secon d s
ℏ
⊙
= (1secon d )KE
e
K E
m
K E
p
(Planet Da y) =
ℏ
2
⋆
GM
3
m
⋅
1
c
ℏ
⋆
K E
p
=
ℏ
2
⋆
GM
3
m
⋅
1
c
1
K E
p
=
ℏ
⋆
GM
3
m
⋅
1
c
K E
p
=
GM
3
m
ℏ
⋆
⋅ c
K E
p
=
1
2
M
p
v
2
p
v
2
p
=
GM
⋆
r
p
1
2
M
p
v
2
p
=
GM
3
m
ℏ
⋆
⋅ c
1
2
M
p
GM
⋆
r
p
=
GM
3
m
ℏ
⋆
⋅ c
1
2
M
p
M
⋆
M
3
m
⋅
1
r
p
=
c
ℏ
⋆
ℏ
2
⋆
= 4c
2
M
6
m
M
2
p
M
2
⋆
r
2
e
ℏ
2
⋆
GM
3
m
⋅
1
c
= 1secon d
4c
G
⋅
M
3
m
M
2
p
M
2
⋆
r
2
p
= 1secon d
M
3
m
M
2
p
⋅
r
2
p
M
2
⋆
=
G
4c
(1secon d )
of 70 74
We know . It could be say the mass of spectral class G2V star. If we know the spectral class of the
star we know its luminosity, and by the inverse square law for luminosity we know , the distance of the
planet in the habitable zone from the star. We just need to choose , the mass of the moon, and ,
the mass of the planet such that we get the 1 second on the right side of the equation to find where all
characteristic times intersect, including that of the proton. So, if we write it!
!
We spent time deriving this equation for the archaeological aspects of this work. If we want to find
answer to the mystery of why the characteristic times are a second, and close to it where that of the
planet day is concerned, we might need to use it. The Planet Day characteristic time for our Sun is one
second, and that of the proton is always one second. The Planet Day characteristic time is close to a
second for the Sun, so if we put in the mass of our moon, the mass of Earth, and the mass of the Sun,
we should get something close to the second on the right. Let’s do that…!
!
And, it does come close showing all of the characteristic times closely meet with our solar system.!
M
⋆
r
p
M
m
M
p
M
3
m
M
2
p
⋅
r
2
p
M
2
⋆
⋅
4c
G
= (1secon d )
(7.347673E 22kg)
3
(5.972E 24kg)
2
⋅
(1.496E11m)
2
(1.9891E 30kg)
2
⋅
4(299,792,458m /s)
(6.674E − 11)
= 1.13secon d s
of 71 74
Appendix 1
If we want to prove that our planetary Planck constant is correct, the delocalization time for the
Earth should be 6 months using it, the time for the Earth to travel the width of its orbit. We want
to solve the Schrödinger wave equation for a wave packet and use the most basic thing we can
which is a Gaussian distribution. The solution is standard. We want to then substitute for
Planck’s constant that is used for quanta and atoms our Planck-type constant (h bar solar)
for the Earth/Moon/Sun system then apply it to predict the delocalization time for the Moon in
its orbit with the Earth around the Sun.
We consider a Gaussian wave-packet at t=0:
We say that is the delocalization length and decompose the wave packet with a Fourier
transform:
is the harmonics of the wave function. We use the identity that gives the integral of a
quadratic:
Solve the equation
With the initial condition
A plane wave is the solution:
Where,
The wave-packet evolves with time as
Calculate the Gaussian integral of
h
ℏ
⊙
ψ (x,0) = Ae
−
x
2
2d
2
d
ψ (x,0) = Ae
−
x
2
2d
2
=
∫
dp
2π ℏ
ϕ
p
e
i
ℏ
px
ϕ
p
∫
∞
−∞
e
−α
2
x+βx
d x =
π
α
e
β
2
4α
iℏ∂
t
ψ (x, t) =
p
2m
ψ (x, t)
ψ (x,0) =
∫
dp ⋅ e
p
2
d
2
2ℏ
2
⋅ e
−
i
ℏ
px
e
i
ℏ
( px−ϵ( p)t)
ϵ(p) =
p
2
2m
ψ (x, t) =
∫
dp ⋅ e
p
2
d
2
2ℏ
2
⋅ e
−
i
ℏ
( px−
p
2
2m
t)
dp
of 72 74
and
The solution is:
where
is the delocalization distance, which for instance could be the width of an atom. is the
delocalization time, the average time for say an electron to traverse the diameter of the atom and
even leave it, to delocalize. If we substitute for our , and say that the delocalization distance
is for the Moon, the width of the Earth orbit, we should get a half a year for the delocalization
time, the time for the Moon and Earth to traverse the diameter of their orbit around the Sun. We
have
Where is the mass of the Moon, and is the orbital radius of the Moon. We have
Now let’s compute a half a year…
(1/2)(365.25)(24)(60)(60)=15778800 seconds
So we see our delocalization time is very close to the half year over which the Earth and Moon
travel from one position to the opposite side of the Sun. The closeness is
Thus we know our is accurate, it continues to function in a theoretical framework. The thing
about this is that it means we can predict the mass of the Moon from the Earth year. In terms of
what we said earlier that the Moon allows for life by creating the seasons, holding the Earth at
its tilt to the Sun, so we don’t go through extreme heat and cold, this suggests the Moon has a
mass that follows from Earth orbit, which is the habitable zone of the Sun, the right distance for
water to exist as liquid, and thus could be as it is for a reason, which means life might be part of
a physical process throughout the Universe, that it unfolds naturally in the evolution of star
systems.
α
2
=
d
2
2ℏ
2
+
it
2mℏ
β =
i x
ℏ
ψ
2
= exp
[
−
x
2
d
2
⋅
1
1 + t
2
/τ
2
]
τ =
m d
2
ℏ
d
τ
ℏ
ℏ
⊙
τ =
m
moon
(2r
moon
)
2
ℏ
⊙
m
moon
r
moon
τ = 4
(7.34767E 22kg)(3.844E 8m)
2
2.8314E 33J ⋅ s
= 15338227secon ds
15338227
15778800
100 = 97.2 %
ℏ
⊙
of 73 74
Appendix 2
Some of the data used in this paper:
(Proton Mass)
(Planck Constant)
(Proton Radius)
(Gravitational Constant)
(light speed)
(Fine Structure Constant)
Earth day=(24)(60)(60)=86,400 seconds. Using the Moon’s orbital velocity at aphelion, and
Earth’s orbital velocity at perihelion we have:
m
P
: 1.67262 × 10
−27
kg
h : 6.62607 × 10
−34
J ⋅ s
r
p
: 0.833 × 10
−15
m
G:6.67408 × 10
−11
N
m
2
kg
2
c : 299,792,458m /s
α : 1/137
q
p
= q
e
= 1.6022E − 19coulom bs
k
e
= 8.988E 9
Nm
2
C
2
K E
moon
=
1
2
(7.347673E 22kg)(966m /s)
2
= 3.428E 28J
K E
earth
=
1
2
(5.972E 24kg)(30,290m /s)
2
= 2.7396E 33J
of 74 74
The Author
